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I was setting out to do point biserial correlation to look at the relationship between a dichotomous variable and a continuous one, however my continuous variable is non-normal.

The two variables are a yes/no answer to a research question and a score on a psychological tool. Is it valid to do a Mann-Whitney U Test instead?

Many thanks

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The Mann-Whitney U test does not require the tested samples to be normally distributed. The null hypothesis for the test is that the two samples are IID values with the same distribution. The test is based on looking at the relative number of pairwise contests between the groups where one sample is larger than the other. The normal distribution only arises in the test as the approximate null distribution of the test statistic.

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If you are looking for correlation and looking to treat your continuous variable as rank-transformed (as Mann-Whitney would), you might consider Kendall correlation or Spearman correlation. These will work in this case because your other variable is dichotomous (as apposed to be nominal with multiple levels). Usually these give both a measure of association (rho or tau, respectively), and a statistical test.

Other measures of association between a nominal variable and an ordinal variable include Freeman's theta, epsilon-squared, and r (= Z/sqrt(N)). These can be used in association with a Mann-Whitney test.

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